Vargas Lane
03/26/2024 · Primary School

2. Suppose \( z^{3}+2 x^{2} y-y^{2} z=2 \) determines a function \( z=z(x, y) \) of \( x \) and \( y \) locally at \( (x, y, z)=(-1,1,-1) \) (a) Find the tangent plane approximation of \( z(x, y) \) at \( (x, y, z)=(-1,1,-1) \) (b) Find the quadratic surface approximation of \( z(x, y) \) at \( (x, y, z)=(-1,1,-1) \)

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(a) The tangent plane approximation of \( z(x, y) \) at \( (x, y, z)=(-1,1,-1) \) is \( z = 2x - 2y + 2 \). (b) The quadratic surface approximation of \( z(x, y) \) at \( (x, y, z)=(-1,1,-1) \) can be derived from the given expression.

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